Representation dimension of $m$-replicated algebras

TL;DR

This paper investigates the representation dimension of m-replicated algebras derived from hereditary algebras, establishing an upper bound of three and a lower bound of m for the dominant dimension.

Contribution

It proves that the representation dimension of m-replicated algebras is at most three and the dominant dimension is at least m, providing bounds for these invariants.

Findings

Representation dimension of A^{(m)} is at most three.

Dominant dimension of A^{(m)} is at least m.

Establishes bounds linking m-replicated algebras to their homological dimensions.

Abstract

Let $A$ be a finite dimensional hereditary algebra over an algebraically closed field and $A^{(m)}$ the $m$-replicated algebra of $A$. We prove that the representation dimension of $A^{(m)}$ is at most three, and that the dominant dimension of $A^{(m)}$ is at least $m$.